The recursive upper-bound conjecture for neighborly boxes

A box in $\mathbb{R}^d$ is an axis-parallel $d$-dimensional cuboid. Two boxes are $k$-neighborly if their intersection has dimension strictly smaller than $d$ but at least $d-k$. Let $n(k,d)$ be the maximum size of a family of pairwise $k$-neighborly boxes in $\mathbb{R}^d$. We use the boundary conventions $n(0,d)=1$ for all $d\geqslant 1$ and $n(k,d)=2^d$ for $k>d$. The recursive upper-bound conjecture. For every $1\leqslant k\leqslant d$,

$$n(k,d)\leqslant n(k-1,d-1)+n(k,d-1).$$

This supposition was proposed as a possible route to the value $\gamma_k=1/k!$ in the asymptotic limit conjecture. Its status is not resolved in the supplied text.